Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{k^2 - 25}{k - 5}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = k$ $ b = \sqrt{25} = -5$ So we can rewrite the expression as: $p = \dfrac{({k} {-5})({k} + {5})} {k - 5} $ We can divide the numerator and denominator by $(k - 5)$ on condition that $k \neq 5$ Therefore $p = k + 5; k \neq 5$